CLASS 8 MATHS · PART 2 · CHAPTER 5 · STUDY NOTES
Tales by Dots and Lines
Understand tales by dots and lines through short explanations, labelled models and original practice. Read one topic, explain it in your own words, then check.
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The mean balances the distances
Add all observations, then divide by how many observations there are.Four dots balance at 5
Stack count (dots)
Data value (units)
- Observations · One dot per observed value; duplicates stack.
- Mean position · Its horizontal position is the arithmetic mean.
Repeated values each count once in the total, so two dots appear at 4. The dashed mean marker is not another observation.
Read the plotted values
| Data value | Stack count |
|---|---|
| 2 | 1 |
| 4 | 1 |
| 4 | 2 |
| 10 | 1 |
| Data value | Stack count |
|---|---|
| 5 | 0 |
| 5 | 2.6 |
- Arithmetic mean
- The sum of all observations divided by their count.
For 2, 4, 4 and 10, the mean is 20/4 = 5. Below 5, the distances total 3 + 1 + 1 = 5; above 5, the distance is also 5.
The mean need not be an observed value or the midpoint of the smallest and largest observations. Here that midpoint is 6, not 5.
Take a moment. What do you think?
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Question
Find the mean of 3, 5, 5 and 11.
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Track the total as well as the count
Adding a new observation changes both the numerator and denominator.Update both parts
- Updated mean
- New total divided by new count after a specified change.
Starting with 2, 4, 4, 10 gives mean 5. Add 10: total becomes 30 and count 5, so mean becomes 6. Adding a value above the old mean raises it.
Adding 3 and 7 together keeps mean 5 because their own mean is 5. Removing an observation above the mean lowers the mean, if observations remain.
- 1
Recover the old total: mean × count.
- 2
Add or subtract the affected value from that total.
- 3
Update the count if an observation is added or removed.
- 4
Divide the new total by the new count.
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Question
Five values have mean 12. Their total has a recording error of +5. What is the corrected mean?
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Shift every value; the mean shifts with it
A change to every observation differs from adding one new observation.Every point shifts three units
Stack count (dots)
Data value (units)
- Observations · One dot per observed value; duplicates stack.
- Mean position · Its horizontal position is the arithmetic mean.
This is the transformed set 2,4,4,10 plus 3 each. Its mean has moved from 5 to 8.
Read the plotted values
| Data value | Stack count |
|---|---|
| 5 | 1 |
| 7 | 1 |
| 7 | 2 |
| 13 | 1 |
| Data value | Stack count |
|---|---|
| 8 | 0 |
| 8 | 2.6 |
- Data transformation
- Applying the same stated rule to every observation.
If every value increases by 3, the mean also increases by 3. The total increases by 3n and dividing by n leaves +3.
If every value is multiplied by k, the mean is also multiplied by k. For example doubling 2,4,4,10 gives 4,8,8,20 with mean 10.
Explain with algebra
Mean of (x₁ + c), …, (xₙ + c) = old mean + cThere are n copies of c in the new total, so dividing by n adds exactly c.
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Question
Every measured length included an extra 2 cm. If the mean was 148 cm, what is the corrected mean?
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The median depends on sorted positions
Sort first. For an even count, average the two middle observations.A large outlier need not move the median
Stack count (dots)
Data value (units)
- Observations · One dot per observed value; duplicates stack.
- Mean position · Its horizontal position is the arithmetic mean.
The mean is 120/5 = 24, but the third sorted observation, 4, is the median. The large horizontal range makes the outlier visible.
Read the plotted values
| Data value | Stack count |
|---|---|
| 2 | 1 |
| 4 | 1 |
| 4 | 2 |
| 10 | 1 |
| 100 | 1 |
| Data value | Stack count |
|---|---|
| 24 | 0 |
| 24 | 2.6 |
- Median
- The middle observation in an odd sorted list, or the mean of the two middle observations in an even sorted list.
For 2,4,4,10, the middle values are 4 and 4, so the median is 4. If 100 is added, the new median remains 4.
Adding a value above the median cannot lower it, but it does not always strictly increase it: repeated middle values can keep it unchanged. The mean is more sensitive to an extreme value.
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Question
Find the median of 9,2,7,5.
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A frequency tells you how often a value occurs
Do not average only the distinct values when some occur more often.Frequency appears as stacked dots
Stack count (dots)
Data value (units)
- Observations · One dot per observed value; duplicates stack.
- Mean position · Its horizontal position is the arithmetic mean.
There are four observations, not three. Two dots at 3 represent its frequency of two.
Read the plotted values
| Data value | Stack count |
|---|---|
| 2 | 1 |
| 3 | 1 |
| 3 | 2 |
| 5 | 1 |
| Data value | Stack count |
|---|---|
| 3.25 | 0 |
| 3.25 | 2.6 |
- Frequency
- The number of observations with a given value.
- Cumulative frequency
- The total count up to and including a value in an ordered frequency table.
| Value | Frequency | Contribution to sum |
|---|---|---|
| 2 | 1 | 2 |
| 3 | 2 | 6 |
| 5 | 1 | 5 |
| Total | 4 | 13 |
For values 2,3,5 with frequencies 1,2,1, the data are 2,3,3,5. Mean = (2×1 + 3×2 + 5×1)/4 = 13/4 = 3.25.
The median is 3, because sorted positions 2 and 3 both contain 3. Cumulative frequencies 1,3,4 locate those middle positions.
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Question
Values 1 and 4 occur 2 and 3 times. Find the mean.
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A formula uses a range of cells
Column letters and row numbers identify cells in a spreadsheet.Read the column and row
- Cell address
- A column letter followed by a row number, such as B2.
- Cell range
- A group from a start cell to an end cell, such as B2:B5.
If B2:B5 contain 6,8,10,12, =SUM(B2:B5) returns 36 and =AVERAGE(B2:B5) returns 9.
Check the chosen range and numeric entries. A blank cell is not the same as a recorded zero; spreadsheet average functions usually ignore blanks.
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Question
Which range names cells C3, C4, C5 and C6?
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Read axes and scale before describing a trend
A line graph links observations in a meaningful order, often time.Four recorded weeks
Attendance (students)
Week (week number)
- Attendance · Fictional observations.
- 124 · (2, 24)
- 218 · (3, 18)
Invented classroom data to practise reading, not a real school record. Points show observed counts; lines connect them in time order.
Read the plotted values
| Week | Attendance |
|---|---|
| 1 | 20 |
| 2 | 24 |
| 3 | 18 |
| 4 | 26 |
- Trend
- A general pattern of change across the displayed observations.
In the fictional graph, attendance is 20,24,18,26 over four weeks. Week 4 has the highest shown count, but this says nothing about weeks outside the graph.
Connecting points guides reading; it does not prove every intermediate value was measured. Steepness comparisons need the same axis scales and equal time intervals.
- 1
Identify each axis, unit and shown time period.
- 2
Read the values using the scale.
- 3
Describe increases, decreases, peaks and flat intervals.
- 4
Separate what the data show from explanations you would need more evidence to test.
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Question
Attendance changes from 24 to 18. What is the change?
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A strip can show how one whole is shared
Use a clear legend and equal lengths for equal time intervals.Four activities fill one day
- Blue
- Sleep, 8 hours.
- Orange
- Study, 6 hours.
- Violet
- Travel, 2 hours.
- Green
- Other activities, 8 hours.
- Infographic
- A visual presentation combining data, symbols and explanation.
- Activity strip
- A bar divided into time intervals to represent a day or other period.
A fictional 24-hour day can be divided into sleep 8 h, study 6 h, travel 2 h and other activities 8 h. The lengths must add to the full 24 h.
A coloured map needs a legend: a colour may mean a percentage, total, difference or index. Do not guess the quantity from colour alone.
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Question
In a strip with 48 half-hour cells, how many cells represent 3 hours?
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An average does not describe every person
Read how data were collected before turning a graph into a claim.Same mean, different individual values
Stack count (dots)
Data value (units)
- Observations · One dot per observed value; duplicates stack.
- Mean position · Its horizontal position is the arithmetic mean.
Two fictional scores show why a mean cannot establish a minimum for every person.
Read the plotted values
| Data value | Stack count |
|---|---|
| 50 | 1 |
| 90 | 1 |
| Data value | Stack count |
|---|---|
| 70 | 0 |
| 70 | 1.6 |
- Aggregate
- A summary of several observations, such as a group average.
An average sleep duration for an age group is not the sleep duration of every person in that group, and it is not automatically medical advice.
A graph can suggest a question without answering its cause. Sample size, missing groups, collection dates, definition of each measurement and axis ranges all affect interpretation.
Let’s look a little closer.
Go deeper: Compare the same measure
Monthly maximum temperature is not daily average temperature. Rainfall amount in millimetres is not the number of rainy days. Objects launched in one year are not the total still in orbit. Read the measurement definition before comparing.
A cropped vertical scale can reveal small differences but can also make them appear large. State the axis range and check the actual numerical change.
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Question
A group’s mean score is 70. Does every person have a score of at least 70?
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